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5d5048e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 | import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationLeafDomains
namespace OAI
/-! # Entropy of the actual balanced leaf counts
Both the selected count and its complementary half count are treated with
the same unit rounding error. This is applied to the four top blocks and
the two local blocks sampled by the representation experiment.
-/
noncomputable section
namespace LowSpaceSubsetSum
def orientedBlockCount (complement : Bool) (l t k : ℕ) (j : Fin l) : ℕ :=
if complement then t - balancedBlockCounts l k j else balancedBlockCounts l k j
def orientedDensity (complement : Bool) (p : ℝ) : ℝ :=
if complement then 1 / 2 - p else p
theorem orientedDensity_mem {p : ℝ} (hp : p ∈ Set.Icc (1 / 6) (1 / 4))
(complement : Bool) : orientedDensity complement p ∈ Set.Icc (1 / 6) (1 / 3) := by
cases complement <;> simp only [orientedDensity, Bool.false_eq_true, ↓reduceIte]
· constructor <;> linarith [hp.1, hp.2]
· constructor <;> linarith [hp.1, hp.2]
theorem orientedBlockCount_error (complement : Bool) (l t k : ℕ)
(hl : l = 2 ∨ l = 4) (ht : 0 < t) (hk : k ≤ l * t) (j : Fin l) :
|(orientedBlockCount complement l t k j : ℝ) -
(2 * t : ℕ) * orientedDensity complement ((k : ℝ) / (l * (2 * t) : ℕ))| ≤ 1 := by
have hl0 : 0 < l := by rcases hl with rfl | rfl <;> decide
have hlR : (l : ℝ) ≠ 0 := by positivity
have htR : (t : ℝ) ≠ 0 := by positivity
have heq : ((2 * t : ℕ) : ℝ) * ((k : ℝ) / (l * (2 * t) : ℕ)) = (k : ℝ) / l := by
push_cast
field_simp [hlR, htR]
have hcount := balancedBlockCounts_le l t k hl hk j
have herr := balancedBlockCounts_error l k hl0 j
cases complement
· simpa only [orientedBlockCount, orientedDensity, Bool.false_eq_true,
↓reduceIte, heq] using herr
· have hsub : ((t - balancedBlockCounts l k j : ℕ) : ℝ) =
(t : ℝ) - balancedBlockCounts l k j := Nat.cast_sub hcount
simp only [orientedBlockCount, orientedDensity, ↓reduceIte, hsub, mul_sub, heq]
have htwo : ((2 * t : ℕ) : ℝ) * (1 / 2) = t := by push_cast; ring
rw [htwo]
rw [show (t : ℝ) - balancedBlockCounts l k j - (t - (k : ℝ) / l) =
-((balancedBlockCounts l k j : ℝ) - (k : ℝ) / l) by ring, abs_neg]
exact herr
theorem orientedBlockCount_entropy (complement : Bool) (l t k : ℕ)
(hl : l = 2 ∨ l = 4) (ht : 24 ≤ 2 * t) (hk : k ≤ l * t)
(hp : (k : ℝ) / (l * (2 * t) : ℕ) ∈ Set.Icc (1 / 6) (1 / 4)) (j : Fin l) :
((2 * t : ℕ) : ℝ) *
binaryEntropy ((orientedBlockCount complement l t k j : ℝ) / (2 * t : ℕ)) ≤
((2 * t : ℕ) : ℝ) *
binaryEntropy (orientedDensity complement ((k : ℝ) / (l * (2 * t) : ℕ))) + 3 := by
exact rounded_leaf_component ht (orientedDensity_mem hp complement)
(orientedBlockCount_error complement l t k hl (by omega) hk j)
namespace RepresentationSetup
variable {n : ℕ} (R : RepresentationSetup n)
theorem guess_density (g : Guesses n) (hm : 0 < mixerSize n) (j : Fin 3) :
((g j).val : ℝ) / mixerSize n ∈ Set.Icc (1 / 6) (1 / 4) := by
have hg := Finset.mem_Icc.mp (g j).property
have hlo : mixerSize n ≤ 6 * (g j).val := by
have h := hg.1
change (mixerSize n + 5) / 6 ≤ (g j).val at h
omega
have hhi : 4 * (g j).val ≤ mixerSize n := by
have h := hg.2
change (g j).val ≤ mixerSize n / 4 at h
omega
have hmR : (0 : ℝ) < mixerSize n := by exact_mod_cast hm
have hloR : (mixerSize n : ℝ) ≤ 6 * (g j).val := by exact_mod_cast hlo
have hhiR : (4 : ℝ) * (g j).val ≤ mixerSize n := by exact_mod_cast hhi
constructor
· apply (le_div_iff₀ hmR).mpr
linarith
· apply (div_le_iff₀ hmR).mpr
linarith
theorem rawLeaf_top_entropy (z : Layout n) (g : Guesses n) (f : R.Partitions z)
(B : Fin n → Fin 2 × Fin 4) (hn : 400 ≤ n) (j : Fin 2) (i : Fin 4) :
((R.rawLeaf z g f B j i).top.card : ℝ) *
binaryEntropy ((R.rawLeaf z g f B j i).topCount /
((R.rawLeaf z g f B j i).top.card : ℝ)) ≤
(mixerSize n : ℝ) * binaryEntropy
(orientedDensity (j != 0) (((g 0).val : ℝ) / mixerSize n)) / 4 + 3 := by
have ht : 24 ≤ 2 * (3 * n / 100) := by omega
have hk : (g 0).val ≤ 4 * (3 * n / 100) := by
have h := (Finset.mem_Icc.mp (g 0).property).2
change (g 0).val ≤ 8 * (3 * n / 100) / 4 at h
omega
have hm : 0 < mixerSize n := by unfold mixerSize; omega
have hp := guess_density g hm 0
have heq : 4 * (2 * (3 * n / 100)) = mixerSize n := by unfold mixerSize; omega
rw [← heq] at hp
have he := orientedBlockCount_entropy (j != 0) 4 (3 * n / 100) (g 0).val
(Or.inr rfl) ht hk hp i
rw [R.rawLeaf_top_card]
have hcount : (R.rawLeaf z g f B j i).topCount =
orientedBlockCount (j != 0) 4 (3 * n / 100) (g 0).val i := by
fin_cases j <;> rfl
rw [hcount]
change ((2 * (3 * n / 100) : ℕ) : ℝ) * _ ≤ _
rw [heq] at he
have hsize : ((mixerSize n : ℕ) : ℝ) = 4 * ((2 * (3 * n / 100) : ℕ) : ℝ) := by
rw [← heq]
push_cast
ring
calc
_ ≤ ((2 * (3 * n / 100) : ℕ) : ℝ) * binaryEntropy
(orientedDensity (j != 0) (((g 0).val : ℝ) / mixerSize n)) + 3 := he
_ = _ := by rw [hsize]; ring
theorem rawLeaf_local_entropy (z : Layout n) (g : Guesses n) (f : R.Partitions z)
(B : Fin n → Fin 2 × Fin 4) (hn : 400 ≤ n) (j : Fin 2) (i : Fin 4) :
((R.rawLeaf z g f B j i).mixer.card : ℝ) *
binaryEntropy ((R.rawLeaf z g f B j i).localCount /
((R.rawLeaf z g f B j i).mixer.card : ℝ)) ≤
(mixerSize n : ℝ) * binaryEntropy
(orientedDensity (decide (2 ≤ i.val)) (((g j.succ).val : ℝ) / mixerSize n)) / 2 + 3 := by
have ht : 24 ≤ 2 * (2 * (3 * n / 100)) := by omega
have hk : (g j.succ).val ≤ 2 * (2 * (3 * n / 100)) := by
have h := (Finset.mem_Icc.mp (g j.succ).property).2
change (g j.succ).val ≤ 8 * (3 * n / 100) / 4 at h
omega
have hm : 0 < mixerSize n := by unfold mixerSize; omega
have hp := guess_density g hm j.succ
have heq : 2 * (2 * (2 * (3 * n / 100))) = mixerSize n := by unfold mixerSize; omega
rw [← heq] at hp
have he := orientedBlockCount_entropy (decide (2 ≤ i.val)) 2 (2 * (3 * n / 100))
(g j.succ).val (Or.inl rfl) ht hk hp ⟨i.val % 2, Nat.mod_lt _ (by decide)⟩
rw [R.rawLeaf_local_card]
have hcount : (R.rawLeaf z g f B j i).localCount =
orientedBlockCount (decide (2 ≤ i.val)) 2 (2 * (3 * n / 100))
(g j.succ).val ⟨i.val % 2, Nat.mod_lt _ (by decide)⟩ := by
fin_cases j <;> fin_cases i <;> rfl
have hhalf : blockHalf n j.succ = 2 * (3 * n / 100) := by fin_cases j <;> rfl
rw [hcount, hhalf]
rw [heq] at he
have hsize : ((mixerSize n : ℕ) : ℝ) = 2 * ((2 * (2 * (3 * n / 100)) : ℕ) : ℝ) := by
rw [← heq]
push_cast
ring
calc
_ ≤ ((2 * (2 * (3 * n / 100)) : ℕ) : ℝ) * binaryEntropy
(orientedDensity (decide (2 ≤ i.val)) (((g j.succ).val : ℝ) / mixerSize n)) + 3 := he
_ = _ := by rw [hsize]; ring
end RepresentationSetup
end LowSpaceSubsetSum
end
end OAI
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